Question

The straight line y=mx+C will be a tangent to the parabola y? - 8x if C =​

Answer 1

SOLUTION

GIVEN

The straight line y = mx + C will be a tangent to the parabola y² = - 8x

TO DETERMINE

The value of C

EVALUATION

Let the straight line y = mx + C is a tangent to the parabola y² = - 8x at (h, k)

Now the equation of the tangent to the parabola y² = - 8x at (h, k) is

$\sf{yk = - 4(x +h) }$

$\sf{ \implies \: yk = - 4x - 4h }$

$\displaystyle\sf{ \implies \: y= - \frac{4x}{k} - \frac{4h}{k} }$

Comparing with y = mx + C we get

$\displaystyle\sf{ \: m= - \frac{4}{k} \: \: \: and \: \: C = - \frac{4h}{k} }$

$\displaystyle\sf{ \implies \: k= - \frac{4}{m} \: \: \: and \: \: h = - \frac{kC}{4} }$

$\displaystyle\sf{ \implies \: k= - \frac{4}{m} \: \: \: and \: \: h = \frac{C}{m} }$

Again (h, k) is a point on the parabola

Thus we get

$\displaystyle\sf{ \implies \: {k}^{2} = - 8h }$

$\displaystyle\sf{ \implies \: \frac{16}{ {m}^{2} } = - \frac{8C}{m} }$

$\displaystyle\sf{ \implies \: C = - \frac{2}{m} }$

FINAL ANSWER

$\boxed{ \: \displaystyle\sf{ \: C = - \frac{2}{m} } \: \: }$

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