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Locus of Focus of Parabolas Touching x=a and y=b | JEE

JEE Maths question with a full step-by-step solution.

Question
The locus of the focus of the family of parabolas having directrix of slope mm and touching the lines x=ax = a and y=by = b is:
Ay+mx=am+by + mx = am + bcorrect
By+mx=am−by + mx = am - b
Cy−mx=am+by - mx = am + b
Dy−mx=am−by - mx = am - b
Solution
Step 1: Let the focus be (h,k)(h, k). The feet of the perpendiculars from it to the tangents x=ax = a and y=by = b are (a,k)(a, k) and (h,b)(h, b), both lying on the tangent at the vertex — a line of slope mm. Step 2: Equating that slope,
b−kh−a=m⇒k+mh=am+b⇒y+mx=am+b.\frac{b - k}{h - a} = m \Rightarrow k + mh = am + b \Rightarrow y + mx = am + b.
 Correct answer: (1)
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