Question Statement

Find the value of $m$ so that the line $3x+4y+m=0$ is tangent to the hyperbola $x^2-y^2-7x-2y+13=0$.

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Background and Explanation

To find the condition for tangency between a line and a curve, we substitute the equation of the line into the equation of the curve to form a single quadratic equation. A line is tangent to a curve if they intersect at exactly one point, which occurs when the discriminant ($D=b^2-4ac$) of the resulting quadratic equation is equal to zero.

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Solution

Given the equations:

1. Line: $3x+4y+m=0$ — (i)
2. Hyperbola: $x^2-y^2-7x-2y+13=0$ — (ii)

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

From equation (i), we isolate $y$: $4y=-3x-m$ $y=\frac{-3x-m}{4}$

Step 2: Substitute $y$ into the hyperbola equation

Substitute the expression for $y$ into equation (ii): $x^2-{\left(\frac{-3x-m}{4}\right)}^2-7x-2\left(\frac{-3x-m}{4}\right)+13=0$

Expanding the terms: $x^2-\frac{9x^2+6mx+m^2}{16}-7x-\frac{-3x-m}{2}+13=0$

Step 3: Simplify to a quadratic equation

Multiply the entire equation by 16 to clear the denominators: $16x^2-(9x^2+6mx+m^2)-112x-8(-3x-m)+208=0$ $16x^2-9x^2-6mx-m^2-112x+24x+8m+208=0$

Group the terms by powers of $x$: $7x^2+(-6m-112+24)x+(-m^2+8m+208)=0$ $7x^2+(-6m-88)x+(-m^2+8m+208)=0$

Factor out a $-2$ from the $x$-coefficient: $7x^2-2(3m+44)x+(-m^2+8m+208)=0$

Step 4: Apply the condition for tangency

The line will be tangent to the hyperbola if the discriminant of the quadratic equation is zero ($b^2-4ac=0$): $[-2(3m+44){]}^2-4(7)(-m^2+8m+208)=0$ $4(3m+44{)}^2-28(-m^2+8m+208)=0$

Divide the entire equation by 4 to simplify: $(3m+44{)}^2-7(-m^2+8m+208)=0$

Expand the terms: $(9m^2+264m+1936)+7m^2-56m-1456=0$ $16m^2+208m+480=0$

Step 5: Solve for $m$

Divide the equation by 16: $m^2+13m+30=0$

Factor the quadratic: $m^2+10m+3m+30=0$ $m(m+10)+3(m+10)=0$ $(m+3)(m+10)=0$

This gives two possible values for $m$: $m+3=0\Rightarrow m=-3$ $m+10=0\Rightarrow m=-10$

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Key Formulas or Methods Used

• Substitution Method: Substituting the linear equation into the conic equation.
• Condition for Tangency: For a quadratic equation $a{x}^2+bx+c=0$, the line is tangent if the discriminant $D=b^2-4ac=0$.
• Quadratic Expansion: $(a+b{)}^2=a^2+2ab+b^2$.
• Factorization: Solving a quadratic equation by splitting the middle term.

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Summary of Steps

1. Rearrange the linear equation to express $y$ in terms of $x$.
2. Substitute this $y$ value into the hyperbola's equation.
3. Multiply by the common denominator to eliminate fractions and simplify the equation into the standard quadratic form $a{x}^2+bx+c=0$.
4. Identify the coefficients $a,b,$ and $c$ and set the discriminant $b^2-4ac$ to zero.
5. Solve the resulting quadratic equation in $m$ to find the required values.