Question Statement

Find the value of $a$ ( $a \neq = 0$ ) so that the line $a x - 2 y + 3 = 0$ is tangent to the parabola:

$y^{2} - 2 y + 3 x + 7 = 0$

Background and Explanation

To find the condition for a line to be tangent to a curve, we substitute the equation of the line into the equation of the curve to form a quadratic equation. For the line to be a tangent, it must touch the curve at exactly one point, meaning the resulting quadratic equation must have exactly one real solution. This occurs when the discriminant ( $D = b^{2} - 4 a c$ ) is equal to zero.

Solution

Given the equation of the line: $a x - 2 y + 3 = 0 \dots (i)$

And the equation of the parabola: $y^{2} - 2 y + 3 x + 7 = 0 \dots (ii)$

Step 1: Express $x$ in terms of $y$

From equation (i), we isolate $x$ : $a x = 2 y - 3$ $x = \frac{2 y - 3}{a}$

Step 2: Substitute $x$ into the parabola equation

Substitute this expression for $x$ into equation (ii): $y^{2} - 2 y + 3 (\frac{2 y - 3}{a}) + 7 = 0$

To clear the fraction, multiply the entire equation by $a$ : $a y^{2} - 2 a y + 3 (2 y - 3) + 7 a = 0$ $a y^{2} - 2 a y + 6 y - 9 + 7 a = 0$

Step 3: Arrange into standard quadratic form

Group the terms involving $y$ to write the equation in the form $A y^{2} + B y + C = 0$ : $a y^{2} + (6 - 2 a) y + (7 a - 9) = 0$ $a y^{2} + 2 (3 - a) y + (7 a - 9) = 0$

Step 4: Apply the tangency condition

The line will be tangent to the parabola if the discriminant of the quadratic equation is $0$ . Using $D = B^{2} - 4 A C = 0$ : $[2 (3 - a)]^{2} - 4 (a) (7 a - 9) = 0$

Expand the terms: $4 (9 - 6 a + a^{2}) - 4 (7 a^{2} - 9 a) = 0$

Divide both sides by $4$ to simplify: $(9 - 6 a + a^{2}) - (7 a^{2} - 9 a) = 0$ $a^{2} - 6 a + 9 - 7 a^{2} + 9 a = 0$ $- 6 a^{2} + 3 a + 9 = 0$

Step 5: Solve for $a$

Divide the equation by $- 3$ : $2 a^{2} - a - 3 = 0$

Factor the quadratic equation: $2 a^{2} - 3 a + 2 a - 3 = 0$ $a (2 a - 3) + 1 (2 a - 3) = 0$ $(a + 1) (2 a - 3) = 0$

Setting each factor to zero:

$a + 1 = 0 \Rightarrow a = - 1$
$2 a - 3 = 0 \Rightarrow a = \frac{3}{2}$

The required values of $a$ are $- 1$ and $\frac{3}{2}$ .

Key Formulas or Methods Used

Substitution Method: Substituting the linear equation into the non-linear equation to find intersection points.
Tangency Condition: For a quadratic equation $A x^{2} + B x + C = 0$ , the line is tangent if the discriminant $D = B^{2} - 4 A C = 0$ .
Quadratic Factorization: Solving for the variable by breaking down the middle term.

Summary of Steps

Isolate $x$ from the line equation $a x - 2 y + 3 = 0$ .
Substitute the expression for $x$ into the parabola equation $y^{2} - 2 y + 3 x + 7 = 0$ .
Simplify the resulting equation into a standard quadratic form in terms of $y$ .
Set the discriminant ( $b^{2} - 4 a c$ ) of this quadratic to zero.
Solve the resulting quadratic equation in $a$ to find the possible values.

Question Statement

Find the value of $a$ ( $a \neq = 0$ ) so that the line $a x - 2 y + 3 = 0$ is tangent to the parabola:

$y^{2} - 2 y + 3 x + 7 = 0$

Background and Explanation

Solution

Given the equation of the line: $a x - 2 y + 3 = 0 \dots (i)$

And the equation of the parabola: $y^{2} - 2 y + 3 x + 7 = 0 \dots (ii)$

Step 1: Express $x$ in terms of $y$

From equation (i), we isolate $x$ : $a x = 2 y - 3$ $x = \frac{2 y - 3}{a}$

Step 2: Substitute $x$ into the parabola equation

Substitute this expression for $x$ into equation (ii): $y^{2} - 2 y + 3 (\frac{2 y - 3}{a}) + 7 = 0$

To clear the fraction, multiply the entire equation by $a$ : $a y^{2} - 2 a y + 3 (2 y - 3) + 7 a = 0$ $a y^{2} - 2 a y + 6 y - 9 + 7 a = 0$

Step 3: Arrange into standard quadratic form

Group the terms involving $y$ to write the equation in the form $A y^{2} + B y + C = 0$ : $a y^{2} + (6 - 2 a) y + (7 a - 9) = 0$ $a y^{2} + 2 (3 - a) y + (7 a - 9) = 0$

Step 4: Apply the tangency condition

The line will be tangent to the parabola if the discriminant of the quadratic equation is $0$ . Using $D = B^{2} - 4 A C = 0$ : $[2 (3 - a)]^{2} - 4 (a) (7 a - 9) = 0$

Expand the terms: $4 (9 - 6 a + a^{2}) - 4 (7 a^{2} - 9 a) = 0$

Divide both sides by $4$ to simplify: $(9 - 6 a + a^{2}) - (7 a^{2} - 9 a) = 0$ $a^{2} - 6 a + 9 - 7 a^{2} + 9 a = 0$ $- 6 a^{2} + 3 a + 9 = 0$

Step 5: Solve for $a$

Divide the equation by $- 3$ : $2 a^{2} - a - 3 = 0$

Factor the quadratic equation: $2 a^{2} - 3 a + 2 a - 3 = 0$ $a (2 a - 3) + 1 (2 a - 3) = 0$ $(a + 1) (2 a - 3) = 0$

Setting each factor to zero:

$a + 1 = 0 \Rightarrow a = - 1$
$2 a - 3 = 0 \Rightarrow a = \frac{3}{2}$

The required values of $a$ are $- 1$ and $\frac{3}{2}$ .

Key Formulas or Methods Used

Substitution Method: Substituting the linear equation into the non-linear equation to find intersection points.
Tangency Condition: For a quadratic equation $A x^{2} + B x + C = 0$ , the line is tangent if the discriminant $D = B^{2} - 4 A C = 0$ .
Quadratic Factorization: Solving for the variable by breaking down the middle term.

Summary of Steps

Isolate $x$ from the line equation $a x - 2 y + 3 = 0$ .
Substitute the expression for $x$ into the parabola equation $y^{2} - 2 y + 3 x + 7 = 0$ .
Simplify the resulting equation into a standard quadratic form in terms of $y$ .
Set the discriminant ( $b^{2} - 4 a c$ ) of this quadratic to zero.
Solve the resulting quadratic equation in $a$ to find the possible values.

7.5 Q-3

Question Statement

Background and Explanation

Solution

Step 1: Express $x$ in terms of $y$

Step 2: Substitute $x$ into the parabola equation

Step 3: Arrange into standard quadratic form

Step 4: Apply the tangency condition

Step 5: Solve for $a$

Key Formulas or Methods Used

Summary of Steps

7.5 Q-3

Question Statement

Background and Explanation

Solution

Step 1: Express $x$ in terms of $y$

Step 2: Substitute $x$ into the parabola equation

Step 3: Arrange into standard quadratic form

Step 4: Apply the tangency condition

Step 5: Solve for $a$

Key Formulas or Methods Used

Summary of Steps